Not particularly, personally I would say the Amish are not living in an orthodox expression of Christianity. Anything that focuses that much on the law is missing the entire point of what Christ did.
Aside from "dressing differently" than others, I don't see the connection.
Amish are Anabaptists who eschew technology and Chassidim are groups within Judaism that followed the Baal Shem Tov (directly or indirectly through his disciples.) His focus was on serving G-d with Joy and a focus on mysticism and Kabbalah for everyone in the community.
I can't say I am expert enough in Amish culture and theology to comment on their similarity. Does anyone else see some similarity (aside from perhaps perceiving them as "other" and "orthodox?"
Nope. They are the break-off of the Mennonites, one of the earliest of the Anabaptist protestant denominations. They have their own set of rules that are not necessarily tied to scripture.
I do not believe there is any Christian denomination that is equivalent (in the approach to religion) to Ultra-orthodox Judaism. There may be groups that sequester themselves away from general society among Catholics and Orthodox Christians in order to follow the rigid orthodoxy, but they would be limited to monasteries, etc.
No. You are comparing apples to oranges. Both are fruits, but they are each unique in their own way.
Amish are a branch of the Anabaptist denominations. Anabaptism came about as a response to the unbiblical views of a lot of state churches during the early Reformation in the late 1500's. They did not see Luther or other reformers as going far enough to root out unbiblical practices and interpretations prevalent at the time. They were not trying to be legalistic, just more closely follow what the Bible says. Their culture is carefully guarded to try to avoid becoming influenced by, entangled with, or dependent upon the world around them.
For more information about Anabaptist history, get a copy of Cup and Cross: An Introduction to Anabaptist History by Michael S. Martin.
They are similar in that they dress and sort of talk the same, and both are tight-knit communities away from mainstream. Both are observant according to their definitions and religious, although different religions that are different more than the new testament.. Both have fashions and hats from 18th Century Poland and the Black Forest. Both have German dialects. (Only one is part Herbrew- YIddish). While both have ideas of piety and modesty in dress, the rules are different. Amish don't use buttons because that is considered haughty and reminiscent of the military. Hasidim do use buttons. Both wear a lot of black, but so do Italians from the Old Country. Amish are rural mostly farmers, Hasidim are urban dwellers mostly. Amish don't many go to college, Hasidim are usually well educated. Amish don't believe in using new technology. They think calculators and radios are 'of the devil'. Hasidim do use innovations and technology, within their parameters of observance.